3.1022 \(\int \frac{1}{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{2 \text{Unintegrable}\left (\frac{1}{x^2 \sqrt{a^2 c x^2+c} \sqrt{\tan ^{-1}(a x)}},x\right )}{a}-\frac{2 \sqrt{a^2 c x^2+c}}{a c x \sqrt{\tan ^{-1}(a x)}} \]

[Out]

(-2*Sqrt[c + a^2*c*x^2])/(a*c*x*Sqrt[ArcTan[a*x]]) - (2*Unintegrable[1/(x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*
x]]), x])/a

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Rubi [A]  time = 0.210974, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)),x]

[Out]

(-2*Sqrt[c + a^2*c*x^2])/(a*c*x*Sqrt[ArcTan[a*x]]) - (2*Defer[Int][1/(x^2*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]
]), x])/a

Rubi steps

\begin{align*} \int \frac{1}{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}} \, dx &=-\frac{2 \sqrt{c+a^2 c x^2}}{a c x \sqrt{\tan ^{-1}(a x)}}-\frac{2 \int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \sqrt{\tan ^{-1}(a x)}} \, dx}{a}\\ \end{align*}

Mathematica [A]  time = 5.14239, size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)),x]

[Out]

Integrate[1/(x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^(3/2)), x]

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Maple [A]  time = 1.042, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \left ( \arctan \left ( ax \right ) \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{a}^{2}c{x}^{2}+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(1/2),x)

[Out]

int(1/x/arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(1/2),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/atan(a*x)**(3/2)/(a**2*c*x**2+c)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a^{2} c x^{2} + c} x \arctan \left (a x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/arctan(a*x)^(3/2)/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(a^2*c*x^2 + c)*x*arctan(a*x)^(3/2)), x)